Reflective Property of the Parabola

Reflective property. Light, sound, or radio waves originating at the focus of a parabola are reflected parallel to the axis (Fig. 5.60); conversely, waves arriving parallel to the axis are all directed towards the focus (Fig. 5.59). This single property explains every parabola application in §5.7.1: a headlight bulb at the focus produces a parallel beam; a receiver at the focus of a dish concentrates parallel incoming signals.

Worked applications (Examples 5.35–5.37). For $y=\frac1{32}x^2$ (a solar-mirror cross-section), rewriting as $x^2=32y$ gives $4a=32\Rightarrow a=8$, so the heating tube (placed at the focus) sits 8 units above the vertex. For a search-light bowl $40\,\text{cm}$ wide and $30\,\text{cm}$ deep (so $(30,20)$ lies on $y^2=4ax$, taking depth as the $x$-direction): $20^2=4a(30)\Rightarrow4a=\frac{400}{30}=\frac{40}3$, so the bulb sits $a=\dfrac{10}3\,\text{cm}$ from the vertex. For an elliptical-lens system' …